What is a Photon? Foundations of Quantum Field Theory. C. G. Torre

Transcription

1 What is a Photon? Foundations of Quantum Field Theory C. G. Torre May 1, 2018

2 2 What is a Photon? Foundations of Quantum Field Theory Version 1.0 Copyright c Charles Torre, Utah State University. PDF created May 1, 2018

3 Contents 1 Introduction Why do we need this course? Why do we need quantum fields? Problems The Harmonic Oscillator Classical mechanics: Lagrangian, Hamiltonian, and equations of motion Classical mechanics: coupled oscillations The postulates of quantum mechanics The quantum oscillator Energy spectrum Position, momentum, and their continuous spectra Position Momentum General formalism Time evolution Coherent States Problems Tensor Products and Identical Particles Definition of the tensor product Observables and the tensor product Symmetric and antisymmetric tensors. Identical particles Symmetrization and anti-symmetrization for any number of particles Problems Fock Space Definitions Occupation numbers. Creation and annihilation operators Observables. Field operators particle observables particle observables Field operators and wave functions Time evolution of the field operators General formalism

4 4 CONTENTS 4.6 Relation to the Hilbert space of quantum normal modes Problems Electromagnetic Fields Maxwell equations The basic structure of the Maxwell equations Continuity equation and conservation of electric charge The electromagnetic wave equation Electromagnetic energy, momentum, and angular momentum Energy Momentum and angular momentum Electromagnetic potentials Role of the sources Solution to the source-free Maxwell equations Energy and momentum, revisited Angular momentum, revisited Periodic boundary conditions Motion of a charged particle in an electromagnetic field Problems What is a Photon? Hilbert space for the electromagnetic field The Hamiltonian. Stationary states Momentum and Helicity Interpretation in terms of photons Time evolution The electromagnetic field operator Vacuum fluctuations Coherent states Photon interference Limitations on the particle interpretation Problems Spontaneous Emission Hydrogenic atoms The hydrogenic atom coupled to photons Perturbation theory. Fermi s Golden Rule Spontaneous emission First-order approximation Electric dipole transitions Lifetime of 2P 1S Another perspective Problems Epilogue 103 Bibliography 105

5 Chapter 1 Introduction This is a brief, informal, and relatively low-level course in the foundations of quantum field theory. The prerequisites are undergraduate courses in quantum mechanics and electromagnetism. 1.1 Why do we need this course? I have always been dismayed by the fact that one can get a Ph.D. in physics yet never be exposed to the theory of the photon. To be sure, we talk about photons all the time and we know some of their salient properties. But how do they arise in the framework of quantum theory? What do they have to do with the more familiar theory of electromagnetism? Of course, the reason most people don t ever learn what a photon really is is because a photon is an entity which arises in quantum field theory, which is not a physics class most people get to take. Quantum field theory typically arises in most physics curricula either in an advanced quantum mechanics course where one wants to do some many body theory, in a course on quantum optics, or in a course meant to explain the theory underlying high energy particle physics. Of course these subjects can be a bit daunting for someone who just wants to know what it is they are talking about when they use the term photon. But the theory of the photon is not that complicated. The new ingredient is that of a quantum field, which can be explained in some detail to anyone who has a had a decent course in quantum mechanics. The principal goal of this course is really just to explain what are the main ideas behind the quantum field this quantum stuff out of which everything is made. Constructing the quantum field in the context of electromagnetism leads immediately to the notion of the photon. 1.2 Why do we need quantum fields? Let me remind you of the phenomenon of spontaneous emission, in which an atomic electron in an excited state will spontaneously emit one (or more) photons and end up in a lower energy state. This phenomenon will take place in the absence of any external stimulus and cannot be explained using the usual quantum mechanical model of the atom. Indeed, the usual quantum mechanical model of atomic energy levels represents them as stationary states. If an atomic electron occupying an atomic energy level were truly in a stationary state there could never be any spontaneous emission 5

6 6 CHAPTER 1. INTRODUCTION since a stationary state has no time dependent behavior. The only way out of this conundrum is to suppose that atomic energy levels are not really stationary states once you take into account the interaction of photons and electrons. So now we need a quantum theory of more than one particle. We know how to do this, of course. But there is a wrinkle. Think again about the emission of a photon by an atomic electron. The initial state of the system has an electron. The final state of the system has an electron and a photon. Moreover, it is possible to have atomic transitions in which more than one photon appears/disappears. In the usual quantum mechanical formalism the number of particles is fixed. Indeed, the normalization of the wave function for a particle (or for particles) can be viewed as saying that the particle (or particles) is (or are) always somewhere. Clearly we will not be able to describe such processes using the standard quantum mechanical models. If this isn t enough, I remind you that there exist situations in which a photon may disappear, producing an electron-positron pair and, conversely, in which an electron-positron can turn into a photon. So even electrons and positrons are not immune from the appearance/disappearance phenomena. Remarkably, it is possible to describe these variable-particle processes using the axioms of quantum theory, provided these axioms are used in a clever enough way. This new and improved use of quantum mechanics is usually called quantum field theory since it can be viewed as an application of the basic axioms of quantum mechanics to continuous systems (field theories) rather than mechanical systems. The picture that emerges is that the building blocks of matter and its interactions consist of neither particles nor waves, but a new kind of entity: a quantum field. Quantum mechanical particles and their wave-particle duality then become particularly simple manifestations of the quantum field. Every type of elementary particle is described by a quantum field (although the groupings by type depend upon the sophistication of the model). There is an electron field, a photon field, a neutrino field, a quark field and so forth. 1 Quantum field theory (QFT) has led to spectacular successes in describing the behavior of a wide variety of atomic and subatomic phenomena. The success is not just qualitative; some of the most precise measurements known involve minute properties of the spectra of atoms. These properties are predicted via quantum field theory and, so far, the agreement with experiment is perfect. Here I will give a brief, superficial introduction to some of the fundamental ideas underlying QFT. My principal goal will be just to show how QFT is used to describe photons and spontaneous emission. My strategy is to echo Dirac s original constructions, presented in a very influential monograph in 1930 [1], which was aimed at doing just this, and eventually led to modern quantum field theory. In doing so I have borrowed from Merzbacher s [2] and Sakurai s [3] treatment of some of these ideas, which are very clear and closely aligned with the concepts I wanted to introduce to you. It takes a bit more work to use the same ideas to describe, say, electrons in terms of quantum fields. Yet more work is needed to analyze the interaction of these particles (better: these fields). Quantum field theory is a very rich subject, still a subject of intense research. 1.3 Problems 1. What is the experimental value for the mean lifetime of the 2p state of hydrogen? 2. What is positronium? What is its mean lifetime? 1 Relativistic fields include the anti-particles as well.

7 Chapter 2 The Harmonic Oscillator The harmonic oscillator describes motion near stable equilibrium. Its quantum mechanical manifestation is precisely what is needed to understand free (non-interacting) fields and their particle excitations. So we must spend some time establishing the facts we will need concerning the quantum oscillator. This will also give me a chance to review with you the principal parts of quantum mechanics that we will need to define quantum field theory. 2.1 Classical mechanics: Lagrangian, Hamiltonian, and equations of motion Before reviewing the quantum oscillator, it is good to first get oriented by reviewing its classical limit. A harmonic oscillator is characterized by 2 parameters: its mass m and its angular frequency ω. The Lagrangian and Hamiltonian for a (classical) harmonic oscillator are given respectively by L(x, ẋ) = 1 2 mẋ2 1 2 mω2 x 2, H(x, p) = p2 2m mω2 x 2. (2.1) Here x is the displacement from equilibrium, ẋ is the velocity, and p is the momentum canonically conjugate to x. The Euler-Lagrange equations are The Hamilton equations are ẋ = H p, L x d L dt ẋ = 0 ẍ + ω2 x = 0. (2.2) ṗ = H x ẋ = p m, ṗ = mω2 x. (2.3) You can verify that the Lagrangian and Hamiltonian equations of motion are equivalent. The general solution to the equations of motion is given by x(t) = A cos(ωt) + B sin(ωt), (2.4) where A = x(0), B = 1 ẋ(0). (2.5) ω 7

8 8 CHAPTER 2. THE HARMONIC OSCILLATOR A very useful representation of the solutions to the oscillator equation involves complex amplitudes. Define mω a = 2 (x + i mω mω p), a = 2 (x i p). (2.6) mω The general solution to the equation of motion takes the form a(t) = a(0)e iωt, a (t) = a (0)e iωt. (2.7) The Hamiltonian is also the energy of the oscillator. It is conserved, that is, unchanging in time: H(t) = p(t)2 2m mω2 x(t) 2 = p(0)2 2m mω2 x(0) 2 = H(0). (2.8) In terms of the complex amplitudes this result is particularly simple: H = ωa a (2.9) H(t) = ωa (t)a(t) = ω ( a (0)e iωt) ( a(0)e iωt) = ωa (0)a(0) = H(0). (2.10) 2.2 Classical mechanics: coupled oscillations When a dynamical system has more than one degree of freedom, motion near equilibrium is in the form of coupled harmonic oscillations. Let us see how this happens. Suppose the degrees of freedom are denoted by q i, i = 1, 2,..., n, and that the Lagrangian is of the form L = 1 2 g ij(q) q i q j V (q). (2.11) I am using Einstein s summation convention; there is a double sum in the first term of L. The metric g ij (q) is an array which may depend upon the configuration coordinates. There is no loss of generality in assuming the metric to be symmetric, g ij = g ji, (2.12) since only the symmetric combination appears in the sum over i and j. The metric involves the masses and other parameters; it is often just a collection of constants, but need not be. The metric is also often diagonal (e.g., in spherical polar coordinates), but it need not be. The function V represents the potential energy of interaction among the degrees of freedom of the system and with the environment. Critical points of V, i.e., points q0 i such that ( ) V q i (q 0 ) = 0, (2.13) define equilibrium configurations of the system. We will suppose that the equilibrium is stable, i.e., q i 0 is a (local) minimum of V. Let us approximate the motion in the neighborhood of a point of equilibrium by defining y i = q i q i 0, (2.14)

9 2.2. CLASSICAL MECHANICS: COUPLED OSCILLATIONS 9 and expanding the Lagrangian in a Taylor series about y i = 0. To the first non-trivial order we get where L 1 2 M ijẏ i ẏ j 1 2 K ijy i y j L 2 (2.15) M ij = g ij (q 0 ), K ij = ( 2 ) V q i q j (q 0 ), (2.16) and I have dropped an irrelevant additive constant V (q 0 ), i.e., the zero point of potential energy has been chosen to be where q i = q0. i The array M is symmetric, M ij = M ji, and we assume that the potential energy is sufficiently smooth so that the matrix of second partial derivatives is symmetric at the critical point q 0 : K ij = K ji. (2.17) In this approximation, the Euler-Lagrange equations are M ij ÿ j + K ij y j = 0, (2.18) which are a coupled system of n homogeneous, linear ODEs with constant coefficients. Defining y as a column vector with entries y i, and viewing M ij and K ij as symmetric matrices M and K, we can write the EL equations in the matrix form: M y = K y. (2.19) Let us note that if q i 0 is a point of stable equilibrium then the symmetric matrix K is nonnegative, that is, it can have only non-negative eigenvalues. 1 This is because a negative eigenvalue will correspond to displacements y i which lower the potential energy in an arbitrarily small neighborhood of the equilibrium point, which contradicts our assumption of stable equilibrium. If the eigenvalues are positive then the point q i 0 is a local minimum. All this means that q i 0 is a point of stable equilibrium if the quadratic form K( y) := K ij y i y j (2.20) is positive, that is, K( y) > 0 for all y 0. Physically this means that any displacement y i from equilibrium will increase the potential energy. All this discussion is just restating standard results from multivariate calculus. Likewise, positivity of the kinetic energy implies that the symmetric matrix M should be positive definite. This means the quadratic form M( v) := M ij v i v j (2.21) is positive definite, i.e., M( v) > 0 for all v 0. I now would like to appeal to a nice result from linear algebra: If the quadratic form defined by M is positive definite, then there exists a linear change of coordinates y i x i : x i = Λ i jy j, ẋ i = Λ i jẏ j, (2.22) with inverse y i = (Λ 1 ) i jx j, ẏ i = (Λ 1 ) i jẋ j. (2.23) 1 Note that a symmetric, real matrix always admits complete set of eigenvectors with real eigenvalues.

10 10 CHAPTER 2. THE HARMONIC OSCILLATOR such that in the new coordinates x i the approximate Lagrangian takes the form L 2 = 1 2 (ẋ2 1 + ẋ ẋ 2 n) 1 2 (ω2 1x ω 2 2x ω 2 nx 2 n), (2.24) where ω i = ω 1, ω 2,..., ω n are constants. In general these constants can be real, pure imaginary, or zero, depending on whether the eigenvalues of the matrix K are positive, negative, or zero, respectively. This result is called simultaneous diagonalization of quadratic forms. The new coordinates x i given in (2.22) are called the normal modes of vibration. If the quadratic form K is non-negative then ω i 0, and the constants ω i are called the characteristic frequencies. I remind you that under a point transformation such as in (2.22), (2.23) the Euler-Lagrange equations for the new Lagrangian are the original Euler-Lagrange equations expressed in the new coordinates. The new form of the Lagrangian shows that near stable equilibrium the normal modes x i oscillate independently and harmonically at the associated characteristic frequency ω i. To summarize: motion near stable equilibrium of any Lagrangian dynamical system can be viewed as a combination of independent simple harmonic oscillations in one dimension. This means that once you understand a single harmonic oscillator in one dimension (see the previous section) you, in principle, understand any system near stable equilibrium. In the next section we will begin studying the quantum theory of a single harmonic oscillator. The result above suggests that the quantum behavior of more complicated systems near stable equilibrium can be described by a collection of such quantum oscillators, where the displacements correspond to the normal modes of vibration. This point of view leads to a successful description of a variety of physical phenomena, e.g., vibrational spectra of molecules and phonons in solids. It is also one way of describing photons and, more generally, particle excitations of quantum fields. 2.3 The postulates of quantum mechanics In the remainder of this chapter we will review the quantum theory of a simple harmonic oscillator. To begin, I should remind you what it means to have a quantum theory of some physical system. Here is a summary of the axioms of quantum mechanics. The state of a given physical system the values of its observable properties at a given time is represented by a unit vector in a (complex) Hilbert space H. I will call this vector the state vector. Recall that a Hilbert space is a vector space with an inner product which is Cauchy complete in the norm defined by that inner product. The vectors are denoted ψ, the corresponding dual vectors defined by the inner product are denoted ψ, and the Hermitian (or sesquilinear ) inner product of ψ and φ is denoted by φ ψ. This scalar product is linear in ψ and anti-linear in φ. Any observable property A of a physical system is represented by a self-adjoint operator  on H. The possible values that A can take are the (generalized) eigenvalues of Â. The eigenvectors of  represent states in which the corresponding eigenvalue will be measured with statistical certainty. More generally, the probability (density) of getting the (generalized) eigenvalue a in the state ψ is given by a ψ 2, where a is the (generalized) eigenvector of  with (generalized) eigenvalue a. If the eigenvalue is degenerate so that there is more than one (generalized) eigenvector with the eigenvalue a then the probability is obtained by summing (integrating) a ψ 2 over the eigenspace. I will explain all this parenthetical generalized

11 2.4. THE QUANTUM OSCILLATOR 11 stuff in due course. For those who known about such things, I am allowing for self-adjoint operators with a continuous part to their spectra. Self-adjoint operators are used because their (generalized) eigenvectors will form a basis of H. This is needed to make the probability interpretation work. Notice that any two state vectors differing by a phase, ψ and e iα ψ, α R, have the same physical meaning. Time evolution of a physical system is represented by a 1-parameter family of state vectors ψ(t) defined by a self-adjoint operator, the Hamiltonian Ĥ, via the Schrödinger equation: Ĥ ψ(t) = i d ψ(t). (2.25) dt Given a splitting between the quantum system and its (classical) environment, a measurement of the the observable A of the quantum system by a measuring device in the environment with outcome a leaves the quantum system in the (generalized) state represented by a. If the eigenvalue is degenerate, then one gets a (generalized) vector in the degenerate subspace. If you are a seasoned quantum mechanic, these axioms will be old friends. If you are still getting proficient with quantum mechanics, the following illustration of the axioms via the quantum oscillator will hopefully help to get you where you need to be for this course. 2.4 The quantum oscillator The Hilbert space for a harmonic oscillator can be chosen to be the set of complex functions on the real line which are square-integrable 2 ψ ψ(x), dx ψ(x) 2 <. (2.26) Here I am using the symbol to mean corresponds to. As you may know there are many equivalent ways to represent a given state vector. For example, one could just as well define H as the set of complex functions of ψ(x) whose Fourier transform ψ(k) is square integrable and then identify ψ(k) ψ. Following Dirac s approach to quantum mechanics, different representations of the vectors are viewed as the expression of the vectors in different bases for H. It is worth mentioning that the definition of the Hilbert space of square-integrable functions requires us to identify any function whose absolute-value-squared integrates to zero with the zero vector. A Hilbert space has a scalar product. The scalar product of ψ and φ is defined by φ ψ = States of the oscillator are then going to be identified with unit vectors: to us. 1 = ψ ψ = dx φ (x)ψ(x). (2.27) dx ψ(x) 2. (2.28) 2 The integration being used should be understood as Lebesgue integration, but this fine point won t matter much

12 12 CHAPTER 2. THE HARMONIC OSCILLATOR The linear operators ˆx and ˆp corresponding to the position 3 and momentum observables are defined by ˆx ψ x ψ(x), ˆp ψ dψ i dx. (2.29) Here is Planck s constant divided by 2π. I certainly will never prove such things here, but it is an important fact that these linear operators are self-adjoint, as required by the postulates. A necessary condition for this self-adjoint property is that the operators ˆx and ˆp are symmetric 4 with respect to the inner product. An operator  is symmetric if it is equal to its adjoint (on a suitable domain in H). Recall that the adjoint of an operator  is the operator  which satisfies φ  ψ = ψ  φ (2.30) for all vectors φ and ψ. A self-adjoint operator must satisfy  = Â, that is, φ  ψ = ψ  φ. (2.31) Let us check that ˆx is symmetric: ( φ ˆx ψ = dx φ (x)xψ(x) = dx ψ (x)xφ(x)) = ψ ˆx φ. (2.32) It will be useful to note the commutator of the position and momentum operators. We have (on a suitable domain in H) [ˆx, ˆp] ˆxˆp ˆpˆx = i ˆ1, (2.33) where ˆ1 is the identity operator. 2.5 Energy spectrum It turns out that the spectra of both ˆx and ˆp are continuous, so they have generalized eigenvectors and eigenvalues. I will defer the discussion of this until a bit later to keep things from getting too complicated too quickly. Instead, let me first focus on another observable: the Hamiltonian, or energy, H. The operator representing this observable is Ĥ = 1 2m ˆp mω2ˆx 2, (2.34) where I am freely using the fact that one can add linear operators and multiply linear operators by scalars to create new linear operators. It can be shown that Ĥ is self-adjoint and that Ĥ has a purely discrete spectrum. The eigenvalue problem for Ĥ is to solve Ĥ ψ = E ψ 2 2m u (x) mω2 x 2 u(x) = E u(x) (2.35) for a constant E and (normalizable) u(x). The solution of the differential equation (2.35) can be found in any quantum mechanics text. It turns out that the eigenvalues are given by E n = (n + 1 ) ω, n = 0, 1, 2,... (2.36) 2 3 Here position means displacement from equilibrium. 4 In the physics literature symmetric is, unfortunately, usually called Hermitian. To make matters worse, mathematicians often use Hermitian to refer to bounded, self-adjoint operators.

13 2.5. ENERGY SPECTRUM 13 The quantum number n is the number of quanta of energy ω the oscillator has relative to its lowest energy state or ground state. The corresponding eigenvectors correspond to functions which are polynomials in x times a Gaussian in x. Explicitly: E n u n (x) = 1 2n n! where H n (ξ) is the Hermite polynomial in ξ, H n (ξ) = ( 1) n e ξ2 Examples of Hermite polynomials are The energy eigenvectors are orthonormal: ( mω ) 1 4 e mω 2 x2 H n ( π dn 2 dξ n e ξ mω x), (2.37), n = 0, 1, 2,... (2.38) H 0 (ξ) = 1, H 1 (ξ) = 2ξ, H 2 (ξ) = 4ξ 2 2. (2.39) E n E m = δ nm. (2.40) Because Ĥ is self-adjoint its eigenvectors form an orthonormal basis for the Hilbert space H. This means that any element of the Hilbert space can be written as a superposition of energy eigenvectors, that is, there exist complex constants c n such that 5 ψ = c n E n. (2.41) n=0 Here c n = E n ψ are the components of ψ in the basis of energy eigenvectors. Alternatively, we have the operator identity E k E k = ˆ1, (2.42) so that ( ) ψ = E k E k ψ = k k k E k E k ψ = k The normalization of ψ, coupled with the orthonormality of the basis E n, means ψ ψ = c k E k. (2.43) c n 2 = 1. (2.44) n=0 In light of (2.41), knowing the sequence of complex numbers {c 0, c 1, c 2,... } is the same as knowing the state vector ψ. Using the energy eigenvectors the Hilbert space of square integrable function can be identified with the set of square-summable sequences of complex numbers. According to the rules of quantum mechanics, we can interpret c n 2 as the probability that the energy E n is measured when the oscillator is in the state given by ψ. The normalization condition 5 The infinite series converges to ψ in the sense that the difference of ψ and the sequence of partial sums from the right hand side define a vector with zero norm.

14 14 CHAPTER 2. THE HARMONIC OSCILLATOR (2.44) then guarantees that the probabilities for all possible outcomes of an energy measurement add up to one. The complex amplitudes for the oscillator can be viewed as a classical limit of operators which have an important meaning relative to the energy spectrum. These operators are â and â, defined by â = mω 2 (ˆx + i mω mω ˆp), â = 2 (ˆx i ˆp). (2.45) mω Note that I have inserted a convenient factor of 1 into the definition of each operator relative to its classical counterpart. This redefinition makes the amplitudes dimensionless. Because ˆx and ˆp are self-adjoint, you can see from inspection that a and a are adjoints of each other. Using the commutation relations between position and momentum it is easy to see that [â, â ] = ˆ1 (2.46) and Ĥ = ω(â â + 1 2ˆ1). (2.47) The operator 1 2 ωˆ1 represents a simple shift of the zero point of energy from zero. One can drop this term if so desired; this amounts to measuring the energy of the oscillator relative to its ground state energy. The operator ˆN â â represents the energy quanta observable; it is also called the number operator. I say this since ˆN E n = n E n. (2.48) Notice in particular that the ground state satisfies ˆN E 0 = 0, = E 0 â â E 0 = 0. (2.49) The second equation says that the norm of the vector â E 0 vanishes, which also implies the first equation. Therefore the ground state is characterized by the condition More generally, it is easy to compute: â E 0 = 0. (2.50) [ ˆN, â] = â, [ ˆN, â ] = â. (2.51) This implies that â E n E n 1 and â E n E n+1. Explicit computation reveals â E n = n E n 1, â E n = n + 1 E n+1, n = 1, 2,... (2.52) For this reason the operators â, â are sometimes called ladder operators, or annihilation and creation operators of energy quanta. These operators, mathematically speaking, add and subtract energy quanta from energy eigenstates. It is possible to derive the energy spectrum including the energy eigenfunctions just using the commutation relation of ˆN, â, and â. The algebra of the ladder operators more or less defines the quantum oscillator.

15 2.6. POSITION, MOMENTUM, AND THEIR CONTINUOUS SPECTRA Position, momentum, and their continuous spectra The mathematical model of operators representing observables with continuous spectra, e.g., the position and momentum operators, is a little more complicated than that for operators representing observables with discrete spectrum such as the oscillator energy. Operators with discrete spectrum are in many ways like ordinary matrices, albeit with an infinity of matrix elements. Operators with continuous spectrum require some new technology. Let us begin by seeing the difficulties which arise when trying to solve the eigenvalue problem for operators such as position and momentum Position The eigenvalue problem for the operator representing position should be ˆx y = y y xψ y (x) = yψ y (x), (2.53) where y is some (hopefully real) number. No non-trivial function can satisfy this condition. Indeed, if the function is non-zero for at least two values of x there can be no solution. But if the function is non-zero for only one value of x then it is (equivalent to) the zero vector in the Hilbert space. It is possible to solve this position eigenvalue equation if one allows a more general type of eigenfunction, a generalized function, the Dirac delta function δ(x, y). The solution of the eigenvalue equation is then written ψ y (x) = δ(x, y). (2.54) We shall need some of this technology, so let us digress briefly to review it. Strictly speaking, the Dirac delta function is not a function, but rather a distribution. 6 It is possible to treat the Dirac delta as a limit of suitable elements of the Hilbert space. The limit will not exist in the Hilbert space, but the limit of suitable scalar products will exist, and that is all we shall need since the probability interpretation only needs a way to define the scalar products y ψ for all possible values of y. A standard example of this limiting process is as follows. Let y be some given real number and consider the following 1-parameter family of elements of H: φ ɛ,y φ ɛ,y (x) = 1 2π It is straightforward to compute the integral and find 1 ɛ 1 ɛ dk e ik(x y), ɛ > 0. (2.55) ( φ ɛ,y (x) 2 = sin2 π ɛ (x y)) π 2 (x y) 2, (2.56) from which you can see that the function φ ɛ,y (x) is square integrable; indeed for any ɛ > 0: φ ɛ,y φ ɛ,y = φ ɛ,y (x) 2 = 1 ɛ. (2.57) The eigenfunction of position the delta function arises in the limit as ɛ 0. Although we often state this result by writing δ(x, y) = 1 dk e ik(x y), (2.58) 2π 6 Here, a distribution is a continuous linear function on a dense subspace S H of the Hilbert space, where continuous implies a choice of appropriate appropriate topology on S. I shall generally avoid this more rigorous way of doing things.

16 16 CHAPTER 2. THE HARMONIC OSCILLATOR this equation has to be treated carefully. What it means is that for vectors χ in a suitable subspace S H the limit of φ ɛ,y χ as ɛ 0 is defined. We have φ ɛ,y χ = 1 1 ɛ dx dk e ik(x y) χ(x). (2.59) 2π 1 ɛ For suitably nice functions 7 one can interchange the orders of integration and then take the limit as ɛ 0. After interchanging the order of integration we get φ ɛ,y χ = 1 1 ɛ dk dx e ik(x y) χ(x) = 1 2π 1 ɛ 2π 1 ɛ 1 ɛ dk χ(k)e iky, (2.60) where χ(k) is the Fourier transform of χ(x). As ɛ 0 we get the Fourier representation of the function χ(y): lim φ ɛ,y χ = 1 dk χ(k)e iky = χ(y). (2.61) ɛ 0 2π So, we have lim dx φ ɛ,y(x)χ(x) = χ(y), (2.62) ɛ 0 which is the defining relation for the delta function. If desired, one can opt to avoid all this delta function stuff and always work with a small but non-vanishing ɛ. Then the position will not be defined as a mathematical point, but only in terms of a region of size determined by ɛ, but if ɛ is small enough it will not matter physically. While this is satisfying in that one does not need to introduce distributions, this framework is somewhat cumbersome, of course, and that is why one likes to set ɛ = 0 and use the delta function. The preceding results are usually packaged in various simple notations. If you know what you are doing, these notations are very helpful. If you let the notation substitute for understanding, then eventually there will be trouble. The most common notational slogans are: 1 2π dk e ik(x y) = δ(x, y) = δ(y, x) δ(x y), (2.63) dx δ(x, y)f(x) = f(y), (2.64) f(x)δ(x, y) = f(y)δ(x, y), (2.65) lim φ ɛ,y = y, (2.66) ɛ 0 y δ(x, y), (2.67) ˆx y = y y, (2.68) 7 For our purposes, a class of nice functions would be smooth functions whose absolute value decreases faster than the reciprocal of any polynomial in x as x.

17 2.6. POSITION, MOMENTUM, AND THEIR CONTINUOUS SPECTRA 17 y χ = χ(y), (2.69) dy χ(y) y = χ, (2.70) y z = δ(y, z), (2.71) dx x x = ˆ1. (2.72) These last five formal relations are the most important for us. If we augment our Hilbert space by including generalized functions like the delta function then the eigenvectors of position can be accommodated and they constitute a generalization of an orthonormal basis, now labeled by a continuous variable instead of a discrete variable, with summations over eigenvectors becoming integrals, and with the Dirac delta function replacing the Kronecker delta in the orthonormality relation. It will be useful later to have available the following identities which are satisfied by the energy eigenvectors and position eigenvectors: dx u k(x)u l (x) = dx E k x x E l = E k E l = δ kl, (2.73) u k(x)u k (y) = y E k E k x = y x = δ(x, y). (2.74) k k Finally, I remind you the physical interpretation of the (position) wave function, ψ ψ(x) = x ψ. (2.75) The probability P x (a, b) for finding the position x (a, b) is given by P x (a, b) = b a dx ψ(x) 2. (2.76) We say that ψ(x) 2 is the probability density for position, and that ψ(x) 2 dx is the probability for finding the particle in the infinitesimal interval (x, x + dx) Momentum For the momentum operator we want to solve an eigenvalue problem of the form ˆp p = p p i d dx u p(x) = pu p (x), (2.77) where p is (hopefully) a real constant. It is not too hard to solve this differential equation; we have u p (x) = Ae i px, (2.78)

18 18 CHAPTER 2. THE HARMONIC OSCILLATOR where A is a constant. There are two difficulties here. First of all, the spectrum of the momentum operator should consist of real numbers, but the eigenvalue equation allows p to be complex. Secondly, and perhaps more drastically, whether p is real or complex (assuming A 0) the function u p (x) is not square-integrable it is not an element of the Hilbert space. We are in a similar place as we were with the position operator, and we can proceed in a similar way. Once again we can define the eigenfunction as a generalized function (or distribution) via a limit of elements of H. Fix a real number p and consider a 1-parameter family of vectors in H: µ p,ɛ u p,ɛ (x) = 1 2π e i px e ɛ2 x 2, ɛ > 0. (2.79) It is easy to see that these functions are square-integrable for any ɛ > 0: µ p,ɛ µ p,ɛ = π ɛ. (2.80) Of course, u p,ɛ (x) is not a momentum eigenfunction, but it does approach the generalized eigenfunction (2.78) as ɛ 0. While the limit does not lead to an element of H, scalar products with elements of H do admit a limit. We have µ p,ɛ ψ = 1 2π dx e i px e ɛ2 x 2 ψ(x). (2.81) Evidently, as ɛ 0 the limit of the scalar product is proportional to the Fourier transform ψ of ψ: lim µ p,ɛ ψ = 1 dx e i px ψ(x) = 1 ψ(k), k = p/. (2.82) ɛ 0 2π One usually uses a notation such as φ(p) = 1 ψ(k) and calls φ(p) the momentum wave function. Since knowing the Fourier transform of a function is as good as knowing the function, and since the Fourier transform ψ is square-integrable if and only if ψ is, we could define H as the set of square-integrable momentum wave functions. This is the basis for using momentum space wave functions to do quantum mechanics. As with the technology surrounding generalized position eigenvectors, the preceding results are packaged in various simple notations. The most common are: lim µ p,ɛ = p lim u p,ɛ (x) u p (x) = 1 e i px, (2.83) ɛ 0 ɛ 0 2π ˆp p = p p (2.84) p ψ = 1 ψ(p/ ) = φ(p), (2.85) x p = 1 2π e i px, p x = 1 2π e i px, (2.86) dp φ(p) p = ψ. (2.87)

19 2.6. POSITION, MOMENTUM, AND THEIR CONTINUOUS SPECTRA 19 p p = δ(p, p), (2.88) dp p p = ˆ1. (2.89) There are relations involving the energy eigenvectors analogous to what we had when discussing the position eigenvectors: dp u k(p)u l (p) = dp E k p p E l = E k E l = δ kl, (2.90) u k(p)u k (p ) = p E k E k p = p p = δ(p, p ). (2.91) k k Finally, I remind you the physical interpretation of the momentum wave function, ψ φ(p) = p ψ = 1 ψ(p/ ). (2.92) The probability P p (a, b) for finding the momentum in the range (a, b) is given by General formalism P p (a, b) = b a dp φ(p) 2. (2.93) Let us now create a little formalism to summarize these results and to generalize them to other observables which may have a continuous part to their spectrum. All this formalism can be justified using the spectral theory of self-adjoint operators on Hilbert space. Let  be a self-adjoint operator on a Hilbert space H. Its spectrum can consist of a discrete part,  a n = a n a n, n = 1, 2,..., a n R (2.94) and a continuous part,  λ = λ λ, λ U R. (2.95) The eigenvectors a n are elements of H and can be chosen to be orthonormal a m a n = δ mn. (2.96) The λ are limits of elements of H but are not themselves elements of H. The limits of the scalar products of λ with a (dense) subspace of H will exist. These generalized eigenvectors will be orthonormal in the delta function sense : λ λ = δ(λ, λ). (2.97) The eigenvectors and generalized eigenvectors together form a basis for the Hilbert space in the sense that for any ψ H ψ = ψ n a n + dλ ψ(λ) λ. (2.98) n U

20 20 CHAPTER 2. THE HARMONIC OSCILLATOR Equivalently, ˆ1 = n a n a n + dλ λ λ. (2.99) U Given the spectral decomposition (2.98), if the state vector of the system is ψ, the interpretation of ψ n is that ψ n 2 is the probability the system will be found to have the value a n (in the state a n ) for the observable represented by Â. If the eigenvalue a n is degenerate there is more than one linearly independent eigenvector with eigenvalue a n for  then the total probability for getting a n upon a measurement of A is obtained by summing ψ n 2 over the degenerate subspace. The interpretation of ψ(λ) is that b a dλ ψ(λ) 2 is the probability for measuring A and getting the value λ (a, b). If the generalized eigenvalues are degenerate one must sum/integrate this quantity over the (generalized) eigenspace to get the total probability. Some self-adjoint operators have purely discrete spectrum; the Hamiltonian for the harmonic oscillator is an example. Other operators, like the position and momentum have purely continuous spectrum. Operators also may have both continuous and discrete parts to their spectrum. An example of this would be the Hamiltonian for a hydrogen atom, where the bound states the usual atomic energy levels correspond to the discrete spectrum and the scattering (or ionized) states correspond to the continuous spectrum. 2.7 Time evolution Next we will briefly review the dynamics of a quantum oscillator in the Heisenberg picture. I use the Heisenberg picture since this picture of dynamics is the most immediately accessible in quantum field theory. You should be familiar with the equivalent Schrödinger picture of dynamics, so we can start there. The time evolution of a state vector in the Schrödinger picture is, of course, defined by the Schrödinger equation: Ĥ ψ(t) = i d ψ(t). (2.100) dt Assuming that the Hamiltonian has no explicit time dependence, as is the the case for the harmonic oscillator, the solution to this equation for a given initial state ψ(0) can be expressed as ψ(t) = e i tĥ ψ(0). (2.101) You may be more familiar with an alternative but equivalent form of this solution. If the initial vector is expanded in a basis of energy eigenvectors: ψ(0) = n c n E n, (2.102) then (2.101) takes the form ψ(t) = n c n e i Ent E n. (2.103) In the Schrödinger picture, observables A are constructed as operators built from ˆx and ˆp. We write  = Â(ˆx, ˆp), and the expectation value of A at time t, denoted by A (t) is given by A (t) = ψ(t)  ψ(t). (2.104)

21 2.7. TIME EVOLUTION 21 As you may know, all physical predictions of quantum mechanics can be expressed in terms of expectation values of suitable observables, so (2.104) suffices to provide all dynamical information. The Heisenberg picture of dynamics can be understood as arising from a different organization of terms in the fundamental formula (2.104). We write where I ve defined A (t) = ψ(t) Â ψ(t) = ψ(0) e i tĥâe i tĥ ψ(0) ψ(0) Â(t) ψ(0), (2.105) Â(t) = e i tĥâe i tĥ. (2.106) You can see that using the last equality in (2.105) along with (2.106) amounts to assigning the time dependence to the operator rather than to the state vector. The only thing that matters, physically speaking, is the combination of factors which appears in (2.105), so the organization of the time dependence within this expression is a matter of convenience only. Evidently, we can view the mathematical representation of the time evolution of any observable either as a one parameter family of state vectors with a fixed operator representing the observable, or as a one parameter family of operator representatives of the observable and a fixed state vector. The former organization of the mathematics is the Schrödinger picture and the latter organization is the Heisenberg picture. In the Heisenberg picture the operator Â(t) represents the observable A at time t. You might find the Heisenberg picture is more closely aligned with how you learned to think about dynamics in Newtonian mechanics. There we always speak of the time evolution of observables like position and momentum. In the Heisenberg picture of quantum mechanics we do the same thing with the operators representing the observables. In the Heisenberg picture there is no Schrödinger equation the state vector is the same for all time, and it is determined once and for all by initial conditions. The equations governing time evolution are of the form (2.106). For a system like the harmonic oscillator, all observables are built from ˆx and ˆp, so it suffices to understand ˆx(t) = e i tĥ ˆxe i tĥ, ˆp(t) = e i tĥ ˆpe i tĥ, (2.107) in order to understand the time evolution of the oscillator in the Heisenberg picture. These relations can be expressed as differential equations, just as (2.101) can be expressed as (2.100). You can check that if the definitions of x(t) and p(t) are differentiated with respect to time then the result is d dt ˆx(t) = 1 d [ˆx(t), Ĥ], i dt ˆp(t) = 1 [ˆp(t), Ĥ]. (2.108) i These are the Heisenberg equations of motion. They can be considered the quantum versions of Hamilton s equations. In this analogy, the operators represent the classical position and momentum variables, and the commutator represents the Poisson bracket. To see this a little more clearly, we need to compute the commutators appearing in (2.108). This can be done by noting that for any operator Â(t) we have [Â(t), Ĥ] = [e i tĥâ(0)e i tĥ, Ĥ] = e i tĥ[â(0), Ĥ]e i tĥ = [Â(0), Ĥ](t). (2.109) Applying this to the displacement and momentum for a harmonic oscillator gives [ˆx(t), Ĥ] = i m ˆp(t), [ˆp(t), Ĥ] = i mω2ˆx(t), (2.110)

22 22 CHAPTER 2. THE HARMONIC OSCILLATOR so that d dt ˆx(t) = 1 m ˆp(t), d dt ˆp(t) = mω2ˆx(t), (2.111) which have the same form as Hamilton s equations for a harmonic oscillator. Consequently, these equations are straightforward to solve: ˆx(t) = cos(ωt) ˆx + 1 sin(ωt) ˆp, ˆp(t) = cos(ωt) ˆp mω sin(ωt) ˆx. (2.112) mω Here I have used the identification ˆx(0) = ˆx, ˆp(0) = ˆp. (2.113) As you can see, the Heisenberg operators evolve in time in the same way as their classical counterparts. From the forms of ˆx(t) and ˆp(t) we easily compute the Heisenberg form of the creation and annihilation operators â = â(0), â = â (0). We have mω â(t) = 2 (ˆx(t) + i mω ˆp(t)) = e iωt â (2.114) and â (t) = mω 2 (ˆx(t) i mω ˆp(t)) = eiωt â, (2.115) as you might have anticipated based upon the classical analogs of these formulas. Finally, it is straightforward to calculate the Heisenberg form of the Hamiltonian: or Ĥ(t) = 1 2m ˆp2 (t) mω2ˆx 2 (t) = 1 2m ˆp2 (0) mω2ˆx 2 (0) = 1 2m ˆp mω2ˆx 2, (2.116) Ĥ(t) = ω(â (t)â(t) + 1 2ˆ1) = ω(â (t)â(t) + 1 2ˆ1) = ω(â â + 1 2ˆ1). (2.117) Evidently, the Heisenberg form of the Hamiltonian operator is the same as its Schrödinger form. This is because Ĥ(t) = e i Ĥt Ĥe i Ĥt = Ĥ (2.118) owing to the fact that [Ĥ, f(ĥ)] = 0 (2.119) for any function f of Ĥ. This result, Ĥ(t) = Ĥ(0), is equivalent to conservation of energy for the quantum oscillator. 2.8 Coherent States It is worth mentioning a family of states that is useful for making the connection between the classical and quantum oscillator. These are the coherent states, defined as eigenvectors of the lowering operator â: â z = z z. (2.120)

23 2.9. PROBLEMS 23 Note that â is not symmetric and its eigenvalues are in general complex. Indeed, it can be shown that there is a coherent state associated to any complex number z. Since â is not self-adjoint, the usual issues with generalized eigenvectors and continuous spectrum do not occur; these vectors are in the Hilbert space and can be normalized in the usual way. However, they are not orthogonal for different eigenvalues. The coherent states are over-complete, which means that they span the Hilbert space but they are not all linearly independent. It can be shown that the coherent states have non-zero probabilities for all energies to occur. The coherent states enjoy a number of important properties. The real and imaginary parts of the eigenvalues yield the expectation values of position and momentum: 2 z ˆx z = mω R(z), z ˆp z = 2 mωi(z). (2.121) All of these states are minimum uncertainty states: ( x) 2 = z ˆx 2 z z ˆx z 2 = 2mω, (2.122) ( p) 2 = z ˆp 2 z z ˆp z 2 = mω 2, (2.123) x p = 2. (2.124) Finally, if the oscillator is in a coherent state defined by z at time t = 0, then at time t it is in the coherent state defined by ze iωt. This last fact is very easy to see in the Heisenberg picture. This means that the complex eigenvalue z evolves in time in the same way as the classical complex amplitude a, defined in (5.98). The coherent states can be viewed as the states which are closest to classical in that the position and momentum have the minimum possible uncertainty and their expectation values evolve according to the classical equations of motion (as they must by Ehrenfest s theorem). For macroscopic values of mass and frequency, these states represent classical Newtonian behavior to good accuracy. 2.9 Problems 1. A system with 2 degrees of freedom, labeled x and y, has the following Lagrangian: L = 1 2 m(ẋ2 + ẏ 2 ) α(x 2 + y 2 ) β(x y) 2. Find the point or points of stable equilibrium. Find the normal modes and characteristic frequencies of oscillation about the equilibria. 2. Prove the result quoted in the text concerning simultaneous diagonalization of quadratic forms. Hints: (1) The quadratic form M defining the approximate kinetic energy can be used to define a scalar product ( v, w) M( v, w) = v i w j M ij = v T Mw. (2) Every scalar product allows a linear change of basis to an orthonormal basis e i in which ( e i, e j ) = δ ij. (3) The orthonormal basis is unique up to any change of basis e i O j i e j where O is an orthogonal matrix,

24 24 CHAPTER 2. THE HARMONIC OSCILLATOR O T = O 1. The potential energy quadratic form is defined by a symmetric array K. Under a change of orthonormal basis defined by O the array changes by K O T KO = O 1 KO. (4) Use the linear algebra result that any symmetric array, such as K, can be diagonalized via a similarity transformation by an orthogonal matrix. 3. Show that the momentum (2.29) and Hamiltonian (2.34) are symmetric operators. 4. Show that the time evolution defined by the Schrödinger equation (2.100) preserves the normalization of the state vector: d ψ(t) ψ(t) = 0. dt 5. Show that the linear operations defined in (2.52) satisfy the adjoint relation (2.30). (Hint: It is sufficient to check the relation on a basis.) 6. Using the probability interpretation of the state vector, show that the expectation value the statistical mean of an observable A in a state represented by ψ can be calculated by A = ψ Â ψ. 7. Define the projection operator into the ground state for the harmonic oscillator by ˆP 0 ψ = E 0 ψ E 0. Show that this is a linear operator. Show that this operator is symmetric. Find the eigenvalues and eigenvectors of this operator. Show that the expectation value in the state ψ of P 0 is the probability that an energy measurement in the state represented by ψ results in the ground state energy. (In this fashion all probabilities in quantum mechanics can be reduced to computations of expectation values of suitable observables. ) 8. The momentum representation for quantum mechanics uses the Fourier transform to identify the Hilbert space H with square integrable functions of momentum. In the momentum representation the position and momentum operators are given by ˆpφ(p) = pφ(p), ˆxφ(p) = i d dp φ(p). Show that these operators satisfy the commutation relations (2.33). Express the harmonic oscillator Hamiltonian as an operator on momentum wave functions. Using the known spectrum of this Hamiltonian in the position representation, (2.36) and (2.37), deduce the spectrum of the Hamiltonian in the momentum representation. (Hint: You do not have to take any Fourier transforms.) 9. Find the ground and first excited states of the oscillator in the momentum representation by taking the Fourier transform of the position representation wave functions.